Import Question JSON

Current Question (ID: 20013)

Question:

$\text{If the angular frequency of the given motion } y = \sin(\omega t) + \cos(\omega t) \text{ is } k\omega, \text{ then value of } k \text{ is:}$ $1. \frac{1}{2}$ $2. 1$ $3. 2$ $4. \text{none of these}$

Options:

1. $\frac{1}{2}$
2. $1$
3. $2$
4. $\text{none of these}$

Solution:

$\text{Hint: Express the given function in a single sine function.}$ $\text{Step: Find the value of } k.$ $y = \sqrt{2} \left[ \sin(\omega t) \cdot \cos\left(\frac{\pi}{4}\right) + \cos(\omega t) \sin\left(\frac{\pi}{4}\right) \right]$ $= \sqrt{2} \sin\left(\omega t + \frac{\pi}{4}\right)$ $\text{Angular frequency } = \omega$ $k = 1.$ $\text{Option (2) is correct.}$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}